Problem

Find a set of generators and relations for $S_3$.

Solution

The order of $S_3$ is $3! = 6$.

The elements of $S_3$ are;

$x_1 = 1$
$x_2 = (1\ 2)$
$x_3 = (1\ 3)$
$x_4 = (2\ 3)$
$x_5 = (1\ 2\ 3)$
$x_6 = (1\ 3\ 2)$

For generators, notice;

Therefore, $S_3$ is generated by $x_2$ and $x_3$.

For relations, notice;

$$ x_2^2 = x_3^2 = x_4^2 = x_5^3 = x_6^3 = 1 $$

Or, by replacing $x_2, x_3$ with $a,b$,

$$ a^2 = b^2 = (ab)^3 = (ba)^3 = 1 $$

$$\tag*{$\blacksquare$}$$